For second-order anomalous states with \(\mathbb {Z}\)  topological charge protected by rotation symmetry, only a \(\mathbb {Z}_2\)  factor can be attributed to the bulk topology as an intrinsic boundary signature [1]}, [2]}, [3]}, [4]}.
This factor merely measures the existence of anomalous states but not their number.
To illustrate this, consider a cubic crystal with chiral hinge modes, as depicted in Fig. REF (c), as an example of a second-order topological phase protected by magnetic fourfold rotation symmetry \(C_4 \mathcal {T}\) .
A symmetry-allowed decoration with Chern insulators reverses the propagation direction of the chiral hinge modes, thereby changing their \(\mathbb {Z}\)  topological charge by an even number.
